Question: Find the sum of $-1676 + (-1669) + (-1662) +... + (-115) + (-108)$.
Answer: Getting started We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $7$ greater than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-1676})$ and the last term $(a_n = {-108})$ are given in the question. We need to find $n$ (the number of terms). Step 1: Find $n$ (the number of terms) The sequence increases by $-108 - (-1676) = 1568$ from the first term to the last term. Because the sequence increases by $7$ each time, it takes $\dfrac{1568}{7} = 224$ terms to get from the first term to the last term. We still need to count the first term, so there are $224 + 1 = {225}$ terms in the sequence. In other words, $n = {225}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{225}}&= \dfrac {\left({-1676} + ({-108}) \right)}{2} \cdot {225} \\\\ S_{{225}} &= -892 \left(225\right) \\\\ S_{{225}} &= -200{,}700\end{aligned}$ The answer $ -200{,}700 $